scholarly journals Positive derivations on partially ordered strongly regular rings

1984 ◽  
Vol 37 (2) ◽  
pp. 178-180 ◽  
Author(s):  
D. J. Hansen
Keyword(s):  
Regular Ring ◽  
Additional Property ◽  
Partially Ordered ◽  
Strongly Regular ◽  
Regular Rings

AbstractThe author presents a proof that a partially ordered strongly regular ring S which has the additional property that the square of each member of S is greater than or equal to zero cannot have nontrivial positive derivations.

A note on modules over regular rings

1971 ◽  
Vol 4 (1) ◽  
pp. 57-62 ◽  
Author(s):  
K. M. Rangaswamy ◽  
N. Vanaja
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Free Left ◽  
Von Neumann Regular ◽  
Strongly Regular ◽  
Torsion Free ◽  
Regular Rings

It is shown that a von Neumann regular ring R is left seif-injective if and only if every finitely generated torsion-free left R-module is projective. It is further shown that a countable self-injective strongly regular ring is Artin semi-simple.

scholarly journals Some characterizations of π-regular rings with no infinite trivial subring

1991 ◽  
Vol 34 (1) ◽  
pp. 1-5
Author(s):  
Yasuyuki Hirano
Keyword(s):  
Commutative Ring ◽  
Regular Ring ◽  
Finite Ring ◽  
Periodic Ring ◽  
Strongly Regular ◽  
Regular Rings

It is shown that a ringRis a π-regular ring with no infinite trivial subring if and only ifRis a subdirect sum of a strongly regular ring and a finite ring. Some other characterizations of such a ring are given. Similar result is proved for a periodic ring. As a corollary, it is shown that every δ-ring is a subdirect sum of a Unite ring and a commutative ring. This was conjectured by Putcha and Yaqub.

Generalized Radical Rings

1968 ◽  
Vol 20 ◽  
pp. 88-94 ◽  
Author(s):  
W. Edwin Clark
Keyword(s):  
Regular Ring ◽  
Multiplicative Semigroup ◽  
Radical Ring ◽  
Strongly Regular ◽  
Regular Rings

Let R be a ring. We denote by o the so-called circle composition on R, denned by a o b = a + b — ab for a, b ∊ R. It is well known that this composition is associative and that R is a radical ring in the sense of Jacobson (see 6) if and only if the semigroup (R, o) is a group. We shall say that R is a generalized radical ring if (R, o) is a union of groups. Such rings might equally appropriately be called generalized strongly regular rings, since every strongly regular ring satisfies this property (see Theorem A below). This definition was in fact partially motivated by the observation of Jiang Luh (7) that a ring is strongly regular if and only if its multiplicative semigroup is a union of groups.

scholarly journals A note on strongly regular rings

1964 ◽  
Vol 40 (2) ◽  
pp. 74-75
Author(s):  
Jiang Luh
Keyword(s):  
Strongly Regular ◽  
Regular Rings

scholarly journals Commutative von Neumann regular rings are 1-Gröbner

2021 ◽  
pp. 30-30
Author(s):  
Zoran Petrovic ◽  
Maja Roslavcev
Keyword(s):  
Regular Ring ◽  
Finitely Generated ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Von Neumann Regular Rings ◽  
Von Neumann Regular ◽  
Ring Of Polynomials ◽  
Bner Basis ◽  
Regular Rings

Let R be a commutative von Neumann regular ring. We show that every finitely generated ideal I in the ring of polynomials R[X] has a strong Gr?bner basis. We prove this result using only the defining property of a von Neumann regular ring.

Factor-Correspondences in Regular Rings

1982 ◽  
Vol 34 (1) ◽  
pp. 23-30
Author(s):  
S. K. Berberian
Keyword(s):  
Present Article ◽  
Regular Ring ◽  
Decomposition Theorem ◽  
Matrix Rings ◽  
Basic Properties ◽  
Principal Ideals ◽  
Highly Effective ◽  
Left Factor ◽  
Regular Rings

Factor-correspondences are nothing more than a way of describing isomorphisms between principal ideals in a regular ring. However, due to a remarkable decomposition theorem of M. J. Wonenburger [7, Lemma 1], they have proved to be a highly effective tool in the study of completeness properties in matrix rings over regular rings [7, Theorem 1]. Factor-correspondences also figure in the proof of D. Handelman's theorem that an ℵ0-continuous regular ring is unitregular [4, Theorem 3.2].The aim of the present article is to sharpen the main result in [7] and to re-examine its applications to matrix rings. The basic properties of factor-correspondences are reviewed briefly for the reader's convenience.Throughout, R denotes a regular ring (with unity).Definition 1 (cf. [5, p. 209ff], [7, p. 212]). A right factor-correspondence in R is a right R-isomorphism φ : J → K, where J and K are principal right ideals of R (left factor-correspondences are defined dually).

Tensor Products of Dimension Groups and K0 of Unit-Regular Rings

1986 ◽  
Vol 38 (3) ◽  
pp. 633-658 ◽  
Author(s):  
K. R. Goodearl ◽  
D. E. Handelman
Keyword(s):  
Regular Ring ◽  
Hausdorff Space ◽  
Tensor Products ◽  
Matrix Algebras ◽  
Dimension Group ◽  
Grothendieck Groups ◽  
Dimension Groups ◽  
Direct Limits ◽  
Finite Products ◽  
Regular Rings

We study direct limits of finite products of matrix algebras (i.e., locally matricial algebras), their ordered Grothendieck groups (K0), and their tensor products. Given a dimension group G, a general problem is to determine whether G arises as K0 of a unit-regular ring or even as K0 of a locally matricial algebra. If G is countable, this is well known to be true. Here we provide positive answers in case (a) the cardinality of G is ℵ1, or (b) G is an arbitrary infinite tensor product of the groups considered in (a), or (c) G is the group of all continuous real-valued functions on an arbitrary compact Hausdorff space. In cases (a) and (b), we show that G in fact appears as K0 of a locally matricial algebra. Result (a) is the basis for an example due to de la Harpe and Skandalis of the failure of a determinantal property in a non-separable AF C*-algebra [18, Section 3].

On a Sheaf of Division Rings*

1975 ◽  
Vol 17 (5) ◽  
pp. 727-731
Author(s):  
George Szeto
Keyword(s):  
Principal Ideal ◽  
Regular Ring ◽  
Central Idempotent ◽  
Division Rings ◽  
Maximal Ideals ◽  
Strongly Regular ◽  
Th 1

R. Arens and I. Kaplansky ([1]) call a ring A biregular if every two sided principal ideal of A is generated by a central idempotent and a ring A strongly regular if for any a in A there exists b in A such that a=a2b. In ([1], Sections 2 and 3), a lot of interesting properties of a biregular ring and a strongly regular ring are given. Some more properties can also be found in [3], [5], [8], [9] and [13]. For example, J. Dauns and K. Hofmann ([3]) show that a biregular ring A is isomorphic with the global sections of the sheaf of simple rings A/K where K are maximal ideals of A. The converse is also proved by R. Pierce ([9], Th. 11–1). Moreover, J. Lambek ([5], Th. 1) extends the above representation of a biregular ring to a symmetric module.

Centres of Rank-Metric Completions

1985 ◽  
Vol 37 (6) ◽  
pp. 1134-1148
Author(s):  
David Handelman
Keyword(s):  
Regular Ring ◽  
Rank Function ◽  
Von Neumann Regular Ring ◽  
Von Neumann ◽  
Rank Metric ◽  
Von Neumann Regular Rings ◽  
Completion Process ◽  
Von Neumann Regular ◽  
Metric Topology ◽  
Regular Rings

In this paper, we are primarily concerned with the behaviour of the centre with respect to the completion process for von Neumann regular rings at the pseudo-metric topology induced by a pseudo-rank function.Let R be a (von Neumann) regular ring, and N a pseudo-rank function (all terms left undefined here may be found in [6]). Then N induces a pseudo-metric topology on R, and the completion of R at this pseudo-metric, , is a right and left self-injective regular ring. Let Z( ) denote the centre of whatever ring is in the brackets. We are interested in the map .If R is simple, Z(R) is a field, so is discrete in the topology; yet Goodearl has constructed an example with Z(R) = R and Z(R) = C [5, 2.10]. There is thus no hope of a general density result.

Some Remarks on Regular and Strongly Regular Rings

1975 ◽  
Vol 17 (5) ◽  
pp. 709-712 ◽  
Author(s):  
R. Raphael
Keyword(s):  
Generalized Inverse ◽  
Ring Homomorphisms ◽  
Strongly Regular ◽  
Regular Rings

This article presents some new algebraic and module theoretic characterizations of strongly regular rings. The latter uses Lambek’s notion of symmetry. Strongly regular rings are shown to admit an involution and form an equational category. An example due to Paré shows that the category of regular rings and ring homomorphisms between them is not equational. Remarks on quasiinverses and the generalized inverse of a matrix are included. The author acknowledges support from the NRC (A7752) and improvements from W. Blair received after announcement of the results.

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